Page 124 - Electrical Properties of Materials

106 The band theory of solids

3. At the boundary of an allowed band cos ka = ±1; that is,
nπ
k = , n = 1,2,3 ... (7.11)
a
Looking at a typical energy versus k plot (Fig. 7.5), we can see that the discon-
tinuities in energy occur at the values of k specified above. We shall say more
about this curve, mainly about the discontinuities in energy, but let us see first
what the other models can tell us.

7.3 The Ziman model
This derivation relies somewhat less on mathematics and more on physical
intuition. We may start again with the assertion that the presence of lattice
ions will make the free-electron model untenable—at least under certain
circumstances.
Let us concentrate now on the wave aspect of the electron and look upon a
free electron as a propagating plane wave. Its wave function is then
ikx
ψ k =e . (7.12)
We know that waves (whether X-rays or electron waves) can easily move
across a crystal lattice, but not always. There are exceptions. When individual
reflections add up in phase (see Fig. 7.6), i.e. when
nλ =2a sin θ, n = 1, 2, 3, (7.13)

William Henry Bragg and William the simple plane wave picture is no longer valid.
Lawrence Bragg, father and son, Eqn (7.13) is a well-known relationship (called Bragg reflection)for
received the Nobel Prize in 1915. X-rays and, of course, it is equally applicable to electron waves. So we

a

θ

Fig. 7.6
Geometry of reflection from atomic
planes.